#151324

order	512
nilpotency class	5
isoclinism	isoclinic class of #151322
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12186
commutator, Frattini, cosocle	#408	C₂³
prime radical	#30441	C₂
LCS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,8,14)(3,15,20,41)(4,9,25,35)(5,26,30,12)(6,31,7,13)(10,28,37,17)(11,24,34,21)(16,45,51,64)(18,48,19,47)(22,56,49,46)(23,58,60,33)(27,43,29,42)(32,59,62,57)(36,52,44,54)(38,50,63,53)(39,55,40,61);(2,9)(4,21)(5,27)(6,7)(10,32)(11,13)(12,40)(14,35)(15,45)(16,49)(17,52)(18,19)(22,51)(24,25)(26,39)(28,54)(29,30)(31,34)(33,42)(36,63)(37,62)(38,44)(41,64)(43,58)(46,47)(48,56)(50,53)(55,61);(1,3)(2,10,14,37)(4,22)(5,28,30,17)(6,19)(7,18)(8,20)(9,32,35,62)(11,38,34,63)(12,41)(13,44,31,36)(15,26)(16,24)(21,51)(23,59,60,57)(25,49)(27,54,29,52)(33,46)(39,45)(40,64)(42,47)(43,48)(50,61,53,55)(56,58)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fi,g^a=gh,d^b=df,e^b=eghi,f^b=fi,g^b=gi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58520);